Identification of Time-Variant Modal Parameters
Using Time-Varying Autoregressive with Exogenous
Input and Low-Order Polynomial Function
C. S. Huang
∗, S. L. Hung & W. C. Su
Department of Civil Engineering, National Chiao Tung University, Hsinchu, Taiwan
&
C. L. Wu
National Center for Research on Earthquake Engineering, Taipei, Taiwan
Abstract: This work presents an approach that ac-curately identifies instantaneous modal parameters of a structure using time-varying autoregressive with exoge-nous input (TVARX) model. By developing the equiva-lent relations between the equation of motion of a time-varying structural system and the TVARX model, this work proves that instantaneous modal parameters of a time-varying system can be directly estimated from the TVARX model coefficients established from displace-ment responses. A moving least-squares technique incor-porating polynomial basis functions is adopted to ap-proximate the coefficient functions of the TVARX model. The coefficient functions of the TVARX model are rep-resented by polynomials having time-dependent coeffi-cients, instead of constant coefficients as in traditional basis function expansion approaches, so that only low orders of polynomial basis functions are needed. Nu-merical studies are carried out to investigate the effects of parameters in the proposed approach on accurately determining instantaneous modal parameters. Numerical analyses also demonstrate that the proposed approach is superior to some published techniques (i.e., recursive technique with a forgetting factor, traditional basis func-tion expansion approach, and weighted basis funcfunc-tion expansion approach) in accurately estimating instanta-neous modal parameters of a structure. Finally, the pro-∗To whom correspondence should be addressed. E-mail: cshuang@
mail.nctu.edu.tw.
posed approach is applied to process measured data for a frame specimen subjected to a series of base excitations in shaking table tests. The specimen was damaged dur-ing testdur-ing. The identified instantaneous modal parame-ters are consistent with observed physical phenomena.
1 INTRODUCTION
Time-varying systems have many applications in vari-ous fields. In mechanical and civil engineering, a sys-tem with active control devices (Saleh and Adeli, 1994, 1997, 1998; Adeli and Saleh, 1997, 1998; Adeli and Kim, 2004; Kim and Adeli, 2004, 2005a,b,c,d; Jiang and Adeli, 2008a,b) modifying stiffness or damping is a time-varying system. A structure under damage normally ex-hibits nonlinear dynamic behaviors and time-dependent stiffness and damping (e.g., Adeli and Jiang, 2006; Jiang et al., 2007). Variations in system stiffness and damp-ing over time result in time-varydamp-ing modal parameters of the system. Consequently, determining instantaneous modal parameters of a time-varying system is generally very useful when assessing structural damage in real ap-plications.
The time-varying autoregressive with exogenous in-put (TVARX) model is often utilized to establish an input–output relationship of a time-varying linear sys-tem from its dynamic responses and input forces (e.g.,
C
2009 Computer-Aided Civil and Infrastructure Engineering. DOI: 10.1111/j.1467-8667.2009.00605.x
Loh et al., 2000; Nied´zwiecki, 2000). The recursive least-squares approach is one of the most popular techniques to estimate time-dependent coefficients of the TVARX model. The recursive least-squares approach, which is an online approach, was well described by Ljung (1987). Although the recursive least-squares approach has high computational efficiency when estimating time-varying parameters, the approach shows slow tracking capabil-ity for time-varying coefficients and is highly sensitive to initial conditions. To improve these shortcomings, vari-able forgetting factors (Fortescue et al., 1981; Toplis and Pasupathy, 1988; Leung and So, 2005), covariance matrix resetting (Jiang and Cook, 1992; Park and Jun, 1992), the sliding window technique (Choi and Bien, 1989; Belge and Miller, 2000), and Kalman filter (Loh et al., 2000) have been incorporated into the recursive least-squares approach.
Another common approach for establishing the TVARX model is the basis function expansion ap-proach, which shows excellent capability on tracking co-efficients changing with time. Various basis functions, such as the Fourier series (Marmarelis, 1987), Legendre polynomial (Nied´zwiecki, 1988), Walsh function (Zou et al., 2003), and wavelets (Tsatsanis and Giannakis, 1993; Wei and Billings, 2002; Adeli and Samant, 2000; Karim and Adeli, 2002; Ghosh-Dastidar and Adeli, 2003), were used to describe TVARX model coeffi-cients. Selecting the proper basis functions is a key to the success of this approach. Numerical experiences by Zou et al. (2003) indicated that the Legendre polyno-mial performed well for the coefficients that change smoothly over time, whereas the Walsh functions were good for piecewise stationary time-varying coefficients. The basis function expansion approach often needs a great number of basis functions and always has trou-ble determining how many basis functions should be used. Numerical difficulties are often encountered when a large number of basis functions are utilized, especially for polynomial basis functions. Nied´zwiecki (2000) pro-posed a weighted basis function approach to overcome these problems, but his approach has an inherent draw-back of computational inefficiency in establishing the TVARX model.
Accurately establishing the TVARX model from measured dynamic responses of a structure is inad-equate for assessing structural damage even though most published work on the TVARX model is con-cerned with how to establish the model accurately. Fur-thermore, the TVARX model coefficients do not have physical meanings. A popular approach to assess struc-tural damage is based on the changes of modal pa-rameters of the structure (Moaveni et al., 2008; Car-den and Brownjohn, 2008; He et al., 2008; Li and Wu, 2008; Ni et al., 2008) even though a drop in natural frequency itself may not really result from structural
damage (Clinton et al., 2006). Consequently, this work presents a novel procedure for accurately determin-ing instantaneous modal parameters of a time-varydetermin-ing system.
First, this work describes a novel and efficient ap-proach for constructing a suitable TVARX model from the dynamic responses of a structure. The coeffi-cient functions in the TVARX model are constructed through the moving least-squares technique adapted from the mesh-free finite element method for construct-ing shape functions (Liu, 2003). The proposed approach needs only low-order polynomials to accurately approx-imate TVARX model coefficients and considerably im-proves the computational inefficiency of a weighted ba-sis function approach.
Second, this work verifies the equivalence between the equation of motion for a structure and the TVARX model when displacement, velocity, or acceleration re-sponses are utilized. Then, this work demonstrates, for the first time in publication, that the instantaneous modal parameters of a time-varying system can be esti-mated from TVARX model coefficients established us-ing displacement responses but not usus-ing velocity or ac-celeration responses by a traditional technique typically used for the AR or ARX model for estimating modal parameters.
Numerical simulations are performed to validate the effectiveness of the proposed procedure in estimating instantaneous modal parameters accurately. Numerical studies investigate the effects of parameters in the pro-posed procedure on accurate determination of instan-taneous modal parameters. Numerical studies also in-dicate that the proposed approach is superior to some published techniques (i.e., recursive technique with a forgetting factor, traditional basis function expansion approach, and weighted basis function expansion ap-proach) in providing accurate estimation of instanta-neous modal parameters for a structure. Finally, the proposed procedure is applied to process measured data for a frame specimen, subjected to a series of base ex-citations in shaking table tests. The specimen showed strong nonlinear dynamic behaviors because damage occurred during testing.
2 METHODOLOGY
The time-varying structural system encountered in civil and mechanical engineering can be described by the following equation of motion when a single-degree-of-freedom system is considered:
m(t) ¨x+ c(t) ˙x + k(t)x = f (t) (1) where m, c, and k are system mass, damping co-efficient, and stiffness, respectively; ¨x, ˙x, and x are
acceleration, velocity, and displacement, respectively. The time-dependent material properties of the system m, c, and k are likely due to control devices or imposed damage.
The instantaneous modal parameters of a time-varying system given by Equation (1) are defined,
ωn(t)= 2π fn(t)= k(t) m(t) and ξ(t) = c(t) 2m(t)ωn(t) (2) where ωn(t) and ξ(t) are instantaneous natural
fre-quency and damping ratio, respectively, and are time dependent. These definitions are similar to those of modal parameters for a linear time-invariant (LTI) sys-tem. Consequently, the system given by Equation (1) can also be characterized by its instantaneous modal pa-rameters.
When the output responses and input of a time-varying system are measured, the TVARX model is frequently applied to establish the relationship be-tween measured input and output. The mathematical expression of the TVARX model with the order (I, J), TVARX(I, J) for single-input/output systems (or sys-tems with a single degree of freedom) is
y (t)= I i=1 φi(t)y(t− i) + J j=0 θj(t) f (t− j) + an(t) (3) where y(t− i) and f(t − i) are the measured response, which can be acceleration, velocity or displacement, and input at time t− it, respectively; 1/t is the sampling rate of the measurement;φi(t) andθj(t) are coefficient
functions to be determined in the model; an(t) is the
residual error accommodating the effects of measure-ment noise, modeling errors, and unmeasured distur-bances. The relationship between Equations (1) and (3) is examined in the following section.
A moving least-squares approach (Lancaster and ˇSalkauskas, 1990) is employed to construct the coeffi-cient functions. The TVARX model coefficoeffi-cient func-tions are expanded onto a set of basis funcfunc-tions. Ap-proximation theory of a function states that a function can always be expanded by a complete set of basis func-tions (Watson, 1980). Hence, polynomial basis funcfunc-tions are used here; let
φi(t)= Ni n=0 ¯ai ntn= pTi ai and θj(t)= ¯ Nj n=0 ¯bj ntn= ¯pTjbj (4) where pT i = (1, t, t2, . . . , tNi), ¯pTj = (1, t, t2, . . . , t ¯ Nj), aT
i = ( ¯ai 0, ¯ai 1, ¯ai 2, . . . , ¯ai Ni), b T
j = (¯bj 0, ¯bj 1, ¯bj 2, . . . ,
¯bj ¯Nj); ¯ai nand ¯bj nare the coefficients to be determined.
A weighted least-squares technique is applied to de-termine coefficients ¯ai n and ¯bj n in Equation (4). Let
¯
φi kand ¯θj kdenote the true values ofφi(tk) and θj(tk),
respectively. Vector aiis determined by minimizing the
error function defined by
E(t)= ¯li l=1 W(t, tl) pT i(tl)ai− ¯φil 2 (5)
where W(t, tl) is a weight function that must be positive,
and ¯liis the number of nodal points forφi(t). The nodal
points uniformly distribute along the time domain un-der consiun-deration.
Minimizing E yields ∂ E ∂ai = 0
(6)
Careful arrangement of Equation (6) yields
Ai(t)ai = Qi(t) ¯ϕi (7) where Ai(t)= ¯li l=1 W(t, tl) pi(tl) piT(tl), Qi(t)=[qi 1, qi 2, . . . , qi ¯li]. qil = W(t, tl) pi(tl) and ϕ¯i = ( ¯φi 1, ¯φi 2, . . . , ¯φi ¯li) T (8) The solution for aiin Equation (7) is
ai = A−1i (t) Qi(t) ¯ϕi (9)
Equation (9) indicates that ai is dependent on time.
Notably, the number of nodal points should be much larger than the number of basis functions in pi to
en-sure the existence of A−1i . Substituting Equation (9) into Equation (4) results in
φi(t)= ˜ϕi(t) ¯ϕi (10)
where ˜ϕi(t) is a vector of shape functions for φi(t) in
terms of finite element terminology, and ˜
ϕi(t)= piT(t) A−1i (t) Qi(t) (11)
Similarly,θj(t) can be expressed as
θj(t)= ˜θj(t) ¯ϑj (12) where ¯ ϑj = ( ¯θj 1, ¯θj 2, . . . , ¯θj ˆlj) T, ˜θ j(t)= ¯pTj(t) ¯A−1j (t) ¯Qj(t). ¯ Aj(t)= ˆlj l=1W(t, tl) ¯pj(tl) ¯p T j(tl), ¯Qj(t)= [ ¯qj 1, ¯qj 2, . . . , ¯qj ˆlj] ¯qjl = W(t, tl) ¯pj(tl) (13)
and ˆljis the number of nodal points forθj(t). If the same
for each coefficient function in the TVARX model, each coefficient function has the same shape functions, and the formulation becomes simple. In Equations (10) and (12), ¯ϕi and ¯ϑj are unknown. Notably, the number of
nodal points should be much larger than the number of basis functions in pi.
A least-squares approach is applied to determine ¯ϕi and ¯ϑj by minimizing ¯ E= N ¯n= 1 (an(t¯n))2 (14)
where N is the number of data points to be used in establishing the TVARX model. Recall that an(t)
is the noise variable in the TVARX model. From Equations (3), (11), and (12), one can establish
an(t)= y(t) − ⎛ ⎝I i=1 t,iϕ¯i+ J j=0 t, jϑ¯j ⎞ ⎠ (15) where t,i = y (t − i) ˜ϕi(t) and t, j = f (t − j) ˜θj(t) (16)
Substituting Equations (15) and (16) into Equation (14) and minimizing ¯E with respect to ¯ϕi and ¯ϑj yield
VTY˜ = VTV ˜C (17) where V= ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ t1,1 t1,2 . . . t1,I t1,0 t1,1 · · · t1,J t2,1 t2,2 . . . t2,I t2,0 t2,1 · · · t2,J .. . ... . .. ... ... ... . .. ... tN,1 tN,2 · · · tN,I tN,0 tN,1 · · · tN,J ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (18a) ˜
Y=y(t1) y(t2) · · · y(tN)
T (18b) ˜ C= ¯ ϕT 1 ϕ¯ T 2 · · · ϕ¯TI ϑ¯ T 0 ϑ¯ T 1 · · · ϑ¯ T J T (18c) Unknowns ¯ϕi and ¯ϑj are determined from Equation
(17) by
˜
C= (VTV)−1VTY˜ (19)
Then, coefficient functions φi(t) in the TVARX
model, Equation (3), are derived by substituting Equa-tion (19) into EquaEqua-tion (10).
Many weight functions can be used in the above for-mulation (Lancaster and ˇSalkauskas, 1990; Liu, 2003). In this work, the exponential weight function is applied:
W (tm, tp)= e−((tm−tp)/0.3d)2 |t m− tp|/d ≤ 1 0 |tm− tp|/d > 1 (20)
where d defines the dimension of the domain where W = 0 and is called the support of the weight func-tion. This weight function is usually used in curve fitting (Lancaster and ˇSalkauskas, 1990) or constructing shape functions in the mesh-free method (Liu, 2003).
It is desirable to obtain instantaneous modal pa-rameters of the time-varying system after the TVARX model has been established from measured output and input. Instantaneous modal parameters can be evalu-ated by following the same procedure used to estimate modal parameters for an LTI system from the ARX model (Wang and Fang, 1986; Huang, 2001) because the TVARX model is exactly the same as the ARX model at any instantaneous moment. Consequently, the instan-taneous modal parameters are associated with the poles of
λI− λI−1φ
1− λI−2φ2− · · · − λφI−1 = 0 (21)
Because φj (j = 1, 2, . . . , I) is time dependent, the
polesλ are also functions of time. The values of λ are usually complex and appear in a conjugate pair. Letλk
be a root of Equation (21) and λk = ck + idk. Then,
the corresponding instantaneous natural frequency and damping ratio are determined by
ωk= α2 k+ βk2 and ξk= α k ωk (22) where βk= 1 ttan−1 dk ck and αk= 1 2tln(c 2 k+ dk2) (23)
Notably, the system given by Equation (1) with a sin-gle degree of freedom should have only one instanta-neous natural frequency function. Consequently, I in Equation (21), or Equation (3), should theoretically equal two, and only two roots for Equation (21) exist. However, I in Equation (21) is typically larger than two when the responses or input containing noise are em-ployed to construct the TVARX model.
3 TVARX MODEL AND STRUCTURAL DYNAMIC RESPONSES
This section establishes the theoretical relationship be-tween Equations (1) and (3) and further assesses the va-lidity of the procedure in determining the instantaneous modal parameters shown in the previous section.
According to the central difference approach, Equa-tion (1) can be discretized as
m(t) (t)2 + c(t) 2t x(t+ t)= f (t) − k(t)−2m(t) (t)2 x(t) − m(t) (t)2 − c(t) 2t x(t− t) (24)
The equation is further simplified as x(t)= ˆφ1(t)x(t − t) + ˆφ2x(t− 2t) + ˆθ1(t) f (t− t) (25) where ˆ φ1(t)= − k(t− t) − 2m(t − t)/(t)2 γ (t − t) , ˆ φ2(t)= − m(t− t)/(t)2− c(t − t)/(2t) γ (t − t) , ˆ θ1(t)= 1 γ (t − t) and γ (t − t) = m(t− t) (t)2 + c(t− t) (2t) (26) If velocity responses are utilized to establish the discrete equation of Equation (1), differentiating Equation (1) with respect to t once yields
m(t) ¨v(t) + [ ˙m(t) + c(t)]˙v(t)
+[˙c(t) + k(t)]v(t) + ˙k(t)x(t) = ˙f (t) (27) wherev(t) denotes velocity responses. From Equation (1),
x (t)= 1
k (t)[ f (t)− m(t) ˙v (t) − c (t) v (t)] (28) Substituting Equation (28) into Equation (27) and us-ing the central difference approach lead to
v(t) =φ1(t)v (t − t) + φ2(t)v (t − 2t) + θ0(t) f (t) +θ1(t) f (t− t) + θ2(t) f (t− 2t) (29) where φ1(t)= − k(t− t) − 2m(t − t)/(t)2 γ (t − t) , φ2(t)= − m(t− t)/(t)2−c (t− t)/(2t) γ (t − t) , θ0(t)= 1 (2t)γ (t − t) , θ1(t)= − ˙k(t− t) k (t− t)γ (t − t) , θ2(t)= −1 (2t)γ (t − t) , γ (t − t) = m (t− t) (t)2 + c (t− t) (2t) , c (t)= ˙m(t) + c (t) −m (t) ˙k (t) k (t) , k (t)= k(t) + ˙c (t) −c (t) ˙k (t) k (t) (30)
Similarly, if acceleration responses are employed, one can establish
a(t)= ˜φ1(t)a(t− t) + ˜φ2(t)a(t− 2t) + ˜θ0(t) f (t)
+ ˜θ1(t) f (t− t) + ˜θ2(t) f (t− 2t) (31)
where a(t) denotes acceleration responses, and
˜ φ1(t)= − ˜k(t− t) − 2m(t − t)/(t)2 ˜ γ (t − t) , ˜ φ2(t)= − m(t− t)/(t)2− ˜c (t − t)/(2t) ˜ γ (t − t) , ˜ θ0(t)= 1 ˜ γ (t − t) 1 (t)2 − ˙k(t− t) (2t)k(t − t) − ˙ k(t− t) (2t)k(t˙ − t) ⎞ ⎠ , ¯ θ1(t)= 1 ˜ γ (t − t) × −2 (t)2 − k (t− t) ¨k(t − t) −˙k(t− t)2 (k (t− t))2 + ˙k(t− t) ˙ k (t− t) k (t− t)k (t − t) ⎞ ⎠ , ˜ θ2(t)= 1 ˜ γ (t − t) 1 (t)2 ˙k(t− t) (2t) k(t − t) + ˙ k (t− t) (2t)k (t − t) ⎞ ⎠ , ˜ γ (t − t) = m (t− t) (t)2 + ˜c (t− t) (2t) , ˜c(t)= ˙m(t) +c (t)−m (t) ˙ k(t) k(t) , ˜k(t)=k (t) + ˙c (t)− c (t)k (t)˙ k (t) (32)
Equations (25), (29), and (31) indicate the equiva-lence between the discrete form of the equation of mo-tion and TVARX model when noise variable an(t) does
not exist. Accordingly, when displacement responses are used, (I, J) in Equation (3) should be (2, 1) and θ0(t)= 0. When velocity or acceleration responses are
used, (I, J) in Equation (3) should be (2, 2).
As the relationship between Equations (1) and (3) is now known, attention now turns to the evaluation of instantaneous modal parameters. When displacement responses are used to establish the TVARX model,
Equation (25) should be obtained theoretically. Then, the roots of Equation (21) are
λ1,2= c1± id1 (33) where c1= 2m(t− t) − (t)2k(t− t) 2m(t− t) + c(t − t)t , d1=t 4m(t− t)k(t − t) − (t)2k2(t− t) + c2(t− t)1/2 2m(t− t) + c(t − t)t (34) The argument, t − t, in functions m, c, and k, is not shown in the following equations to simplify the expres-sions. Substituting Equation (34) into Equation (23) yields α1= 1 2tln 2m− ct 2m+ ct and β1= 1 t tan−1 t4mk− k2(t)2+ c21/2 2m− k(t)2 (35)
The Taylor expressions ofα1andβ1given by Equation
(35) are α1= −2πξ T + O t T 2 and β1= (2π)[1 − ξ2]1 2 T + 1 3!T 3πT2 2[1− ξ2]12 + 24π3[1− ξ2]1 2 − 16π3[1− ξ2]3 2 t T 2 + O t T 4 (36)
where T= 2π√m/k, which is the instantaneous pe-riod, and O((t/T)n) are terms with an order in (t/T)
higher than (or equal to) n. It is well known that (t/T) must be sufficiently small to have an accurate finite dif-ference presentation for the equation of motion. Ac-cordingly, when (t/T) approaches zero, Equation (36) can be simplified as α1= − 2πξ T and β1= 2π T (1− ξ 2)1/2 (37)
Substituting Equation (37) into Equation (22) yields
ω1= 2π T = ! k m and ξ1= ξ = c 2√km (38) Similarly, one obtains the following results from the coefficient functions of the TVARX model established
using velocity or acceleration responses:
ω1v= k m and ξ1v = c 2 km (39) or ω1a = ˜k m and ξ1a = ˜c 2√˜km (40) where subscriptsv and a denote the computational re-sults obtained from velocity and acceleration responses, respectively. Accordingly, ω1v, ω1a, ξ1v, and ξ1a
ob-tained from the TVARX model can differ significantly from the true instantaneous modal parameters of a structural system under consideration. Comparison of Equations (1) and (27) reveals that ω1v and ξ1v are
equal to the true instantaneous modal parameters at t = ˆt when ˙m(ˆt), ˙c(ˆt), and ˙k(ˆt) equal zero.
4 NUMERICAL VERIFICATION
Numerical simulation responses were processed to demonstrate the feasibility of the proposed procedure. The Runge–Kutta method was applied to determine the needed dynamic responses of Equation (1), with a time increment (t) equal to 0.001 second. Consider a system with a single degree of freedom defined by Equation (41) subjected to the base excitation (ag(t))
(Figure 1),
¨x (t)+ 2ξ (t) ωn(t) ˙x (t)+ ωn(t)2x (t)= −ag(t) (41)
Two instantaneous modal parameter types are con-sidered here and defined as follows:
Type I: Slowly varying system
ωn(t)= 2π 1.5 − 0.5 sin 2π 60t , ξ(t) = 4 + 2 sin 2π 60t , 0 ≤ t ≤ 30 (42) Type II: Periodically varying system
ωn(t)= 2π 1.0 − 0.5 sin 2π 10t , ξ(t) = 5 + 2.5 sin 2π 10t , 0 ≤ t ≤ 30 (43) The instantaneous modal parameters defined by Equa-tion (43) vary with time much more rapidly than those defined by Equation (42). Displacement re-sponses were processed in the following system iden-tification. Figure 1 presents the time histories of displacement responses for slowly varying and period-ically varying systems.
Fig. 1. Input acceleration and displacement responses.
The effects of noise and some parameters involved in the proposed procedure on determining instantaneous modal parameters are investigated thoroughly. The pa-rameters considered are the support of the weight func-tion (d in Equafunc-tion (20)), the order of polynomial ba-sis (Ni and ¯Nj in Equation (4)), and number of nodal
points (¯liin Equation (5)). For simplicity, Niand ¯Njare
set to equal ¯N for all i and j; ¯li is also set to equal ˆl for
all i.
4.1 Parametric studies
The TVARX(2, 1) model is used in this section. After setting the desired parameters (i.e., d, ¯N, and ˆl), the
Table 1
Means and variances of relative errors of identified instantaneous modal parameters
Slowly varying system Periodically varying system
σ (%) μ(%) σ (%) μ(%) ˆl N¯ d fn ξ fn ξ fn ξ fn ξ 20 0 2 0.645 1.566 1.070 2.187 6.721 11.81 8.091 20.64 4 0.077 0.245 0.141 0.301 1.623 3.224 2.018 4.305 6 0.014 0.022 0.011 0.022 0.469 1.401 0.493 1.406 1 2 0.033 0.104 0.039 0.120 4.145 5.115 2.848 6.553 4 0.011 0.033 0.018 0.047 1.381 2.062 1.462 2.365 6 0.001 0.006 0.007 0.014 0.524 1.587 0.531 1.524 2 2 0.053 2.907 0.037 1.428 1.051 4.293 1.513 5.427 4 0.003 0.008 0.006 0.013 0.770 3.048 1.096 3.655 6 0.001 0.004 0.007 0.013 0.446 1.485 0.652 2.024 3 2 37.57 38.08 28.57 65.52 34.71 63.27 29.53 102.2 4 0.001 0.005 0.007 0.014 0.562 1.736 0.880 2.501 6 0.001 0.005 0.007 0.014 0.414 1.100 0.605 1.832 35 0 2 0.107 0.642 0.171 0.705 1.069 11.59 1.447 10.15 4 0.005 0.029 0.008 0.024 0.104 0.232 0.083 0.177 6 0.002 0.009 0.007 0.014 0.080 0.136 0.068 0.153 1 2 0.007 0.034 0.011 0.043 0.594 1.959 0.601 2.173 4 0.001 0.005 0.007 0.014 0.050 0.108 0.059 0.108 6 0.001 0.005 0.007 0.014 0.023 0.035 0.047 0.067 2 2 0.001 0.008 0.007 0.016 0.214 2.385 0.332 2.480 4 0.001 0.005 0.007 0.014 0.026 0.106 0.049 0.106 6 0.001 0.005 0.007 0.014 0.032 0.076 0.054 0.087 3 2 0.001 0.025 0.007 0.021 0.108 0.675 0.117 0.580 4 0.001 0.005 0.007 0.014 0.033 0.100 0.052 0.096 6 0.001 0.005 0.007 0.014 0.024 0.037 0.046 0.063
instantaneous modal parameters at any instant time t= tican be determined using the proposed procedure. The
relative error of identified instantaneous modal param-eters at t= tiis defined as
ρid(ti)− ρtrue(ti)
ρtrue(ti)
× 100% (44)
where ρtrue and ρid are the true and identified
instan-taneous natural frequency (fn) or damping ratio (ξ),
respectively. Table 1 summarizes the means (μ) and variances (σ ) of relative errors of instantaneous param-eters identified using different numbers of nodal points (ˆl= 20 and 35), various orders of the polynomial ba-sis ( ¯N= 0, 1, 2, and 3), and different supports of the weight function (d= 2, 4, and 6 seconds). Figures 2 and 3 display the means of relative errors of the identified in-stantaneous modal parameters acquired using different numbers of nodal points for slowly varying and periodi-cally varying systems, respectively.
Table 1 and Figures 2 and 3 show several interesting facts. Using the parameter values given in Table 1 yields accurately identified fnorξ, except for some results
ob-tained using d= 2. The instantaneous modal parameters identified using d= 4 or 6 are considerably more accu-rate than those obtained using d= 2, especially in the cases of small ˆl (i.e., ˆl= 20) and large ¯N (i.e., ¯N = 3). Figures 2 and 3 show that using larger ˆl yields more accurate results of identification when d = 4 or 6 and
¯
N= 0, 1, 2, or 3 are utilized. The proposed approach does not need polynomial basis functions with high or-ders. These facts demonstrate the important features of the proposed procedure.
Figures 4 and 5 present the comparison of the iden-tified instantaneous modal parameters with true values for slowly varying and periodically varying systems, re-spectively. The instantaneous modal parameters were identified using d = 4, ¯N = 2, and ˆl = 30 in the pro-posed approach. A recursive technique with a constant forgetting factor equal to 0.95 (Ljung, 1987) was also applied to determine the instantaneous modal param-eters shown in Figures 4 and 5. The results of the pro-posed approach are highly accurate, with the relative er-rors much less than 1%, and substantially better than those obtained using the recursive technique. When the periodically varying system is considered, the recursive
Fig. 2. Means of relative errors of identified fnandξ varying with ˆl for a slowly varying system: (a) d = 4 seconds; (b) d= 6 seconds.
technique has very limited capability of identifying the instantaneous damping ratios varying with time.
Figure 6 depicts the instantaneous parameters for the periodically varying system identified using a traditional basis function expansion technique and a weighted ba-sis function approach (Nied´zwiecki, 2000) whose re-sults are denoted by “BF” and “WBF,” respectively. The “BF” results were obtained using 20 polyno-mial basis functions ({1, t, t2,. . . , t19}), although the
“WBF” results were determined using the same poly-nomial basis functions ({1, t, t2}) and weighting function
(Equation (20) with d= 4) as those employed to obtain
the results denoted by “proposed” in Figure 5. Compar-ison of the results in Figures 5 and 6 reveals that the pro-posed approach gives much more accurate results than the traditional basis function expansion technique and the approach of Nied´zwiecki (2000) do. Notably, using more polynomial basis functions in the traditional basis function expansion approach does not significantly im-prove the accuracy of results. An IntelR-based PC with
2.40-GHz CPU was used to calculate all the numerical results shown here. It took 28.2 and 54.3 seconds of CPU time to obtain the results utilizing the present approach and the approach of Nied´zwiecki (2000), respectively.
Fig. 3. Means of relative errors of identified fnandξ varying with ˆl for a periodically varying system: (a) d = 4 seconds; (b) d= 6 seconds.
4.2 Effects of noise and selection of a suitable order of the TVARX model
Noise is always found in measured data. White noise was added to numerical simulation displacement re-sponses and input acceleration to assess the effect of noise on the accuracy of instantaneous modal param-eters identified using the proposed approach. The vari-ance of the noise-to-signal ratio is set at 5%. Similar to processing noisy data for a time-invariant system, the order of the TVARX model must increase to accom-modate noise.
Figures 7 and 8 depict the means of relative errors of identified instantaneous modal parameters varying with
the orders of the TVARX model for the slowly varying and periodically varying systems, respectively. For sim-plicity, I is set equal to J in the TVARX(I, J) model in the following computations. The instantaneous modal parameters were obtained using ¯N= 2, d = 4 or 6, and ˆl= 25, 30, or 35. The means of relative errors of identi-fied instantaneous modal parameters generally decrease as I and J increase. As expected, noise significantly im-pacts the accuracy of identified instantaneous modal parameters, especially when identifying instantaneous modal damping ratios. The accuracy of the identified re-sults (Figures 7 and 8) using large I and J is not as good as those (Table 1) for data without noise. Nevertheless, Figures 7 and 8 show that using large I and J can yield
Fig. 4. Identified fnandξ for a slowly varying system.
Fig. 6. Identified fnandξ for a periodically varying system by two existing approaches.
Fig. 7. Variation of means of relative errors of identified fnandξ with the orders of the TVARX model for a slowly varying system: (a) d= 4 seconds; (b) d = 6 seconds.
the means of relative errors of identified instantaneous modal frequencies and damping ratios less than 2% and 10%, respectively.
In real applications, the true instantaneous modal pa-rameters are typically unknown. That is, one cannot estimate the means of relative errors of identified in-stantaneous modal parameters and decide the suitable order of the TVARX model. However, the Akaike in-formation criterion (AIC) (Akaike, 1973), which was originally developed for a time-invariant system, is also often used to determine suitable I and J values in the TVRAX(I, J) model (Tsatsanis and Giannakis, 1993). The AIC is defined as
AIC= NTln V+ 2Nd (45)
where NT is the number of data points used to
con-struct the TVARX model, V is the mean square error of the one-step-ahead prediction from the TVARX model, and Ndis the total number of parameters in determining
the coefficients of the TVARX model. In the proposed procedure, Nd = (I + J + 1) × ˆl because each
coeffi-cient function in the TVARX(I, J) model is expanded using a series of functions with ˆl parameters to be de-termined. Figure 9 shows the values of the AIC varying with orders of the TVARX model. These values of the AIC were obtained using ¯N= 2, ˆl = 35, and d = 4 or 6 seconds in the proposed approach. The value of the AIC generally decreases as I and J increase. When I and J are fixed, the values of the AIC determined us-ing d = 4 seconds are significantly smaller than those obtained with d = 6 seconds. This trend is not consis-tent with thatμ of fnandξ obtained using d = 6 may be
smaller than those obtained with d= 4 (Figures 7 and 8). The reason for this inconsistency can be because the AIC value indicates level of agreement between the re-sponses predicted by the TVARX(I, J) model and mea-sured responses. When meamea-sured data contain noise, the TVARX model giving better prediction does not guarantee to yield more accurate identification of modal parameters.
An error index ( ¯μ) similar to the mean of relative errors of identified instantaneous modal parameters (Equation (44)) is proposed to supplement AIC when selecting a suitable order of the TVARX model. The error index is defined as
¯ μ(I, J) = 1 n n k=1 ρ(I, J, tk)− ρ(I − 1, J − 1, tk) ρ(I, J, tk) (46) where ρ (I, J, tk) denotes the identified
instanta-neous natural frequency or damping ratio at t = tk
obtained using the TVARX(I, J) model. When the TVARX(I, J) model results in highly accurate identi-fication of instantaneous modal parameters, the value of ¯μ(I, J) will likely be very small. Figure 9 presents
the ¯μ(I, J) of fn andξ varying with I and J, where fn
andξ were obtained using ¯N = 2, ˆl = 35, and d = 4 or 6 seconds. The values of ¯μ(I, J) of fn and ξ generally
decrease as I and J increase.
The following two criteria are employed when choos-ing a suitable order of the TVARX model:
1. A suitable I and J must be chosen from a range of I and J in which ¯μ(I, J) changes with small fluctu-ations and must be less than the assigned thresh-old values that are case dependent.
2. A suitable I and J must be chosen from a range of I and J in which the value of the AIC changes with small fluctuations, or a suitable I and J yield the minimum value of the AIC in a broad range of I and J.
When d = 4 seconds was used, the TVARX(41, 41) and TVARX(50, 50) models were good models for the slowly varying system and periodically varying system, respectively (Figures 9 and 10). When d= 6 seconds was used, the TVARX(30, 30) and TVARX(40, 40) mod-els were suitable for the slowly varying system and peri-odically varying system, respectively. Figures 11 and 12 display the instantaneous modal parameters identified using these TVARX models for slowly varying and pe-riodically varying systems, respectively. Figures 11 and 12 also present a comparison of the identified instanta-neous modal parameters with true values.
When d = 4 seconds was used, the TVARX(41, 41) model yielded the maximum relative errors of identified fnandξ of 2.2% and 30.2% for the slowly varying
sys-tem, respectively. The TVARX(50, 50) model yields the maximum relative errors of fnandξ of 3.1% and 33.0%,
respectively, for the periodically varying system. Nev-ertheless, most relative errors of ξ are less than 20%. When d = 6 was used, the TVARX(30, 30) model for the slowly varying system yielded 2.1% and 14.5% max-imum relative errors for fnandξ, respectively. Most
rel-ative errors of ξ are less than 10%. The TVARX(40, 40) model for the periodically varying system generates the maximum relative errors of fn and ξ of 3.9% and
22.6%, respectively. These identified fnandξ are
suffi-ciently accurate for damage assessment of a structure because structural damage can be detected with con-fidence as its fundamental frequency shift exceeds 5% (Salawu, 1997).
Figure 13 displays the results identified using the weighted basis function approach proposed by Nied´zwiecki (2000) to process the noisy data for the periodically varying system. The identified instanta-neous fn and ξ were obtained using the same
polyno-mial basis functions ({1, t, t2}) and weighting function
(Equation (20) with d = 6) and TVARX model (TVARX(40, 40)) as those employed to obtain the
Fig. 8. Variation of means of relative errors of identified fnandξ with the orders of the TVARX model for a periodically varying system: (a) d= 4 seconds; (b) d = 6 seconds.
Fig. 9. Values of AIC varying with the orders of the TVARX model: (a) slowly varying system; (b) periodically varying system.
results denoted by “d = 6” in Figure 12. Compar-ing the results in Figure 13 with “d = 6” results in Figure 12 discovers that the present approach gives much better results than the weighted basis function ap-proach in processing noisy data. Furthermore, it took
29.3 and 2,590.7 seconds of CPU time to obtain those results using the present approach and the weighted basis function approach, respectively. The present ap-proach is substantially superior to the weighted ba-sis function approach in accurately and efficiently
Fig. 10. The error index, ¯μ, of fnandξ varying with the orders of the TVARX model: (a) d = 4 seconds; (b) d = 6 seconds.
Fig. 11. Instantaneous modal parameters identified from noisy data for a slowly varying system.
estimating the instantaneous modal parameters of a structure.
5 APPLICATIONS TO MEASURED RESPONSES FROM SHAKING TABLE TESTS
Shaking table tests are vital to understanding the dy-namic behaviors, especially nonlinear behaviors, of
structural systems under earthquakes. The National Center for Research on Earthquake Engineering in Taiwan conducted a series of tests on reinforced con-crete (RC) frames of two columns interconnected by a strong beam to investigate the dynamic behaviors of low-ductility RC columns and to understand their collapse mechanism. The details of testing programs are given by Wu et al. (2006). Figure 14 shows the
Fig. 12. Instantaneous modal parameters identified from noisy data for a periodically varying system.
45 15 17 40 100 40 44 112.5 20 Load Cells g
a
36 160 20 27 Shaking Table (Side View) (Front View) A BFig. 14. A sketch of experiment setup.
dimensions of a typical frame and the test setup. In total, 21 tons of lead ballast was added to the beam (Figure 14) to simulate axial loads on first-story columns in a typical four-story RC building in Taiwan. Ac-celerometers and linear displacement transducers were installed at appropriate locations to measure accelera-tion and displacement responses of a specimen. Load cells were installed between the specimen and shaking table to measure base shear forces.
The specimen was subjected to a series of base excita-tion inputs; that is, it was first shaken under white noise input with small amplitude to estimate its modal param-eters. The test is denoted as “before-damage” test be-cause the specimen was not damaged. Then, the spec-imen was subjected to an earthquake input recorded during the 1999 Chi-Chi earthquake in Taiwan. Strong nonlinear behaviors were observed during this test, and columns near beam connection were damaged. The test is denoted as “during-earthquake” test. Finally, the specimen was shaken again with a low-level white noise input, which is denoted as “after-damage” test. Figure 15 shows the acceleration input and displace-ment response histories during these three tests. Dis-placement shown in Figure 15 is the horizontal rela-tive displacement between A and B (Figure 14). Data
were recorded at a sampling rate of 200 Hz. The Fourier spectra of these displacement responses are given in Figure 16.
The proposed identification procedure with d = 6, ¯
N= 2, and ˆl = 35 was applied to process experimen-tal data (Figure 15). By employing the criteria based on the AIC and ¯μ (discussed in the previous sec-tion), the TVARX(17, 17) and TVARX(34, 34) mod-els were judged good modmod-els for processing data from the “before-damage” and “after-damage” tests, respec-tively, whereas the TVARX(36, 36) model could be appropriate for “during-earthquake” test data. The identified instantaneous modal parameters are shown in Figure 17.
As expected, small variations of instantaneous nat-ural frequencies with time were observed for the cases with white noise input as no further damage occurred under such small input excitation forces. The identified instantaneous natural frequencies in the “before-damage” test are larger than those obtained from the “after-damage” test; the trend is opposite for the identified instantaneous modal damping ratios. The instantaneous natural frequencies in the “before-damage” test are 2.33–2.45 Hz, although those from the “after-damage” test are 1.57–1.66 Hz. These identified
Fig. 15. The input acceleration and response histories from shaking table tests: (a) before damage; (b) during earthquake; (c)
after damage.
Fig. 17. Instantaneous modal parameters identified from
experimental data.
frequencies are close to the peak frequencies observed in the Fourier spectra (Figure 16). The range of identi-fiedξ in the “before-damage” test is 3.3–10% and that forξ from the “after-damage” test is 4.4–16%. As ex-pected, the damaged specimen consumes more damp-ing energy than the specimen before damage when sub-jected to the same level of input acceleration.
The instantaneous natural frequencies from the “during-earthquake” test are close to those identi-fied from the “before-damage” test when t < 2 sec-onds as no damage existed for this duration in the “during-earthquake” test. The value of fn decreased
dramatically around t = 3 seconds, which likely indi-cates specimen damage. As displacement magnitude gradually increases over time to t = 19.15 seconds (Figure 15b), the identified fnvalue generally decreases
to 0.93 Hz andξ increases to 52% from less than 10%. Subsequently, the gradual decrease in magnitude of
Fig. 18. Hysteretic loops between story drift and base shear:
(a) 0< t < 30 seconds; (b) 0 < t < 3.18 seconds. displacement responses leads to an increase in fn and
decrease toξ. These observations obey the well-known physical phenomenon suggesting that structural dam-age decreases the natural frequency and increases the damping ratio of a structure.
To explain further the identified instantaneous modal parameters in the “during-earthquake” test, Figure 18a depicts experimental hysteretic loops between story drift and base shear. The story drift is the relative dis-placement between A and B (Figure 14), and base shear was measured from load cells. The plot was con-structed by using the measured data from t = 0 to 30 seconds. The plot clearly indicates that the speci-men was severely damaged during the test. Figure 18b presents an enlarged section of the plot in Figure 18a for t≤ 3.18 seconds. Notably, the curve slope at an instant
highly depends on structure stiffness at that time. The slopes of the curve between t = 3.02 and 3.05 seconds are significantly smaller than those for t≤ 3.02 seconds (Figure 18b). Hence, a remarkable decrease in fn
ex-ists at roughly t = 3 seconds (Figure 17). Nega-tive slopes exist along the largest hysteretic loop at 19.05≤ t ≤ 19.15 (Figure 18a) and result in the smallest fn(Figure 17). Furthermore, the largest hysteretic loop
indicates the largest energy dissipation in the loop and yields the largest damping ratio (Figure 17).
6 CONCLUDING REMARKS
This work presented a novel approach for accurately estimating instantaneous modal parameters of a time-varying structural system via the TVARX model. A moving least-squares technique with polynomial basis functions—frequently utilized in a mesh-free method for analyzing mechanical problems—is adopted to es-tablish the shape functions for the coefficient functions of the TVARX model. The primary advantages of the proposed approach over a conventional basis function expansion approach and a weighted basis function ap-proach are in using low-order polynomial basis func-tions and in saving computational time, respectively. Finally, the instantaneous natural frequencies and damping ratios are directly estimated from coefficient functions of the TVARX model.
This work theoretically developed the equivalent relations between the equation of motion and the TVARX mode, and further proved that the instanta-neous modal parameters of a time-varying system can be estimated from the TVARX model coefficients es-tablished from displacement responses—but not from velocity or acceleration responses—via a conventional technique typically utilized to identify modal parame-ters of the time-invariant ARX model.
To confirm the validity of the proposed identification approach, numerical simulations of a time-varying sys-tem with a single degree of freedom were performed. This work numerically demonstrated that the proposed approach is much superior to some existing approaches (i.e., recursive technique with a forgetting factor, tradi-tional basis function expansion approach, and weighted basis function expansion approach) in providing accu-rate estimation of instantaneous modal parameters for a structure. Numerical studies indicate that increasing the number of nodal points (ˆl) generally improves the accuracy of identified instantaneous modal parameters. When a large number of nodal points is utilized, the identified instantaneous modal parameters are not sig-nificantly affected by the order of polynomial basis func-tions ( ¯N) and the support of the weight function (d) for
¯
N= 0, 1, 2, 3 and d = 4 or 6 seconds. When data con-taining noise are processed, one must increase the or-der of the TVARX model to identify the instantaneous modal parameters accurately.
To demonstrate the applicability of the proposed ap-proach to real data, responses to shaking table tests were processed. The specimen was subjected to a series of base excitation inputs. The specimen was first shaken under white noise input with small amplitude, then sub-jected to a large earthquake input and damaged, and fi-nally shaken under a small amount of white noise input again. The trend in variations of the identified instanta-neous modal parameters is consistent with the observed physical phenomena during the tests. Changes to instan-taneous modal parameters due to structure damage can be applied to develop useful criteria for assessing dam-age of real structures.
The work concentrated on a system with a single de-gree of freedom or a system with single input/output because the resulting mathematical formulation is sim-ple, its correctness is easily verified, and the parameters controlling the accuracy of the identified instantaneous modal parameters can be studied comprehensively. The formulas given here are easily extended to systems with multiple degrees of freedom. Application of the pro-posed approach to processing measured responses of structures with multiple degrees of freedom and further performing damage assessment of structures are inter-esting and shall be done in the future.
ACKNOWLEDGMENTS
The authors thank the National Science Council of the Republic of China, Taiwan, for financially support-ing this research under Contract No. NSC 94-2625-Z-009-006. The appreciation is also extended to Profes-sor C. H. Loh and the National Center for Research on Earthquake Engineering for providing shaking table test data.
REFERENCES
Adeli, H. & Jiang, X. (2006), Dynamic fuzzy wavelet neu-ral network model for structuneu-ral system identification, Journal of Structural Engineering, ASCE, 132(1), 102– 11.
Adeli, H. & Kim, H. (2004), Wavelet-hybrid feedback least mean square algorithm for robust control of structures, Journal of Structural Engineering, 130(1), 128–37.
Adeli, H. & Saleh, A. (1997), Optimal control of adap-tive/smart bridge structures, Journal of Structural Engineer-ing, 123(2), 218–26.
Adeli, H. & Saleh, A. (1998), Integrated structural/control op-timization of large adaptive/smart structures, International Journal of Solids and Structures, 35(28–29), 3815–30.
Adeli, H. & Samant, A. (2000), An adaptive conjugate gradi-ent neural network—wavelet model for traffic incidgradi-ent de-tection, Computer-Aided Civil and Infrastructure Engineer-ing, 13(4), 251–60.
Akaike, H. (1973), Information theory and an extension of the maximum likelihood principle. 2nd International Sympo-sium on Information Theory, Akademia Kiado, Budapest, 267–81.
Belge, M. & Miller, E. L. (2000), A sliding window RLS-like adaptive algorithm for filtering alpha-stable noise, IEEE Signal Process Letter, 7(4), 86–89.
Carden, E. P. & Brownjohn, J. M. W. (2008), Fuzzy clustering of stability diagrams for vibration-based structural health monitoring, Computer-Aided Civil and Infrastructure Engi-neering, 23(5), 360–72.
Choi, B. Y. & Bien, Z. (1989), Sliding-windowed weighted recursive least-squares method for parameter estimation, Electron Letter, 25(20), 1381–82.
Clinton, J. F., Bradford, S. C., Heaton, T. H. & Favela, J. (2006), The observed wander of the natural frequencies in a structure, Bulletin of the Seismological Society of America,
96(1), 237–57.
Fortescue, T. R., Kershenbaum, L. S. & Ydstie, B. E. (1981), Implementation of self-tuning regulators with variable for-getting factors, Automatica, 17(6), 831–35.
Ghosh-Dastidar, S. & Adeli, H. (2003), Wavelet-clustering-neural network model for freeway incident detection, Computer-Aided Civil and Infrastructure Engineering,
18(5), 325–38.
He, X., Moaveni, B., Conte, J. P., Elgamal, A. & Masri, S. F. (2008), Modal identification study of Vincent Thomas bridge using simulated wind-induced ambient vibration data, Computer-Aided Civil and Infrastructure Engineering,
23(5), 373–88.
Huang, C. S. (2001), Structural identification from ambient vibration measurement using the multivariate AR model, Journal of Sound and Vibration, 241(3), 337–59.
Jiang, J. & Cook, R. (1992), Fast parameter tracking RLS al-gorithm with high noise immunity, Electron Letter, 28(22), 2042–45.
Jiang, X. & Adeli, H. (2008a), Dynamic fuzzy wavelet neu-roemulator for nonlinear control of irregular highrise build-ing structures, International Journal for Numerical Methods in Engineering, 74(7), 1045–66.
Jiang, X. & Adeli, H. (2008b), Neuro-genetic algorithm for nonlinear active control of highrise buildings, International Journal for Numerical Methods in Engineering, 75(7), 770– 86.
Jiang, X., Mahadevan, S. & Adeli, H. (2007), Bayesian wavelet packet denoising for structural system identifica-tion, Structural Control and Health Monitoring, 14, 333– 56.
Karim, A. & Adeli, H. (2002), Incident detection algorithm us-ing wavelet energy representation of traffic patterns, Jour-nal of Transportation Engineering, 128(3), 232–42.
Kim, H. & Adeli, H. (2004), Hybrid feedback-least mean square algorithm for structural control, Journal of Struc-tural Engineering, 130(1), 120–27.
Kim, H. & Adeli, H. (2005a), Hybrid control of smart struc-tures using a novel wavelet-based algorithm, Computer-Aided Civil and Infrastructure Engineering, 20(1), 7–22. Kim, H. & Adeli, H. (2005b), Wavelet hybrid feedback-lms
al-gorithm for robust control of cable-stayed bridges, Journal of Bridge Engineering, 10(2), 116–23.
Kim, H. & Adeli, H. (2005c), Hybrid control of irregular steel highrise building structures under seismic excitations, In-ternational Journal for Numerical Methods in Engineering,
63(12), 1757–74.
Kim, H. & Adeli, H. (2005d), Wind-induced motion control of 76-story benchmark building using the hybrid damper-tuned liquid column damper system, Journal of Structural Engineering, 131(12), 1794–1802.
Lancaster, P. & ˇSalkauskas, K. (1990), Curve and Surface Fitting—An Introduction, Academic Press, NY.
Leung, S. H. & So, C. F. (2005), Gradient-based variable for-getting factor RLS algorithm in time-varying environments, IEEE Transactions on Signal Processing, 53(8), 3141–50. Li, S. & Wu, Z. (2008), A non-baseline method for damage
locating and quantifying in beam-like structure based on dynamic distributed strain measurements, Computer-Aided Civil and Infrastructure Engineering, 23(5), 404–13. Liu, G. R. (2003), Mesh Free Methods: Moving Beyond the
Fi-nite Element Method, CRC Press, NY.
Ljung, L. (1987), System Identification, Theory for the User, Prentice-Hall, Englewood Cliffs, NJ.
Loh, C. H., Lin, C. Y. & Huang, C. C. (2000), Time domain identification of frames under earthquake loadings, Journal of Engineering Mechanics, ASCE, 126(7), 693–703. Marmarelis, V. Z. (1987), Advanced Methods of Physiological
System Modeling, University of Southern California, Los Angeles.
Moaveni, B., He, X., Conte, J. P. & de Callafon, R. A. (2008), Damage identification of a composite beam based on changes of modal parameters, Computer-Aided Civil and Infrastructure Engineering, 23(5), 339–59.
Ni, Y. Q., Zhou, H. F., Chan, K. C., & Ko, J. M. (2008), Modal flexibility analysis of cable-stayed Ting Kau bridge for dam-age identification, Computer-Aided Civil and Infrastructure Engineering, 23(3), 223–36.
Nied´zwiecki, M. (1988), Functional series modelling approach to identification of nonstationary stochastic systems, IEEE Transactions Automatic Control, 33, 955–61.
Nied´zwiecki, M. (2000), Identification of Time-Varying Pro-cesses, John Wiley & Sons, NY.
Park, D. J. & Jun, B. E. (1992), Self-perturbing recursive least squares algorithm with fast tracking capability, Electron Letter, 28(6), 558–59.
Salawu, O. S. (1997), Detection of structural damage through changes in frequency: a review, Engineering Structures, 19, 718–23.
Saleh, A. & Adeli, H. (1994), Parallel algorithms for in-tegrated structural and control optimization, Journal of Aerospace Engineering, 7(3), 297–314.
Saleh, A. & Adeli, H. (1997), Robust parallel algorithms for solution of the Riccati equation, Journal of Aerospace En-gineering, 10(3), 126–33.
Saleh, A. & Adeli, H. (1998), Optimal control of adaptive building structures under blast loading, Mechatronics, 8(8), 821–44.
Toplis, B. & Pasupathy, S. (1988), Tracking improvements in fast RLS algorithm using a variable forgetting factor, IEEE Trans Acoust Speech Signal Process, ASSP, 36(2), 206–27. Tsatsanis, M. K. & Giannakis, G. B. (1993), Time-varying
system identification and model validation using wavelets, IEEE Transactions Signal Processing, 41, 3512–23. Wang, Z. H. & Fang, T. (1986), A time-domain method for
identifying modal parameters, Journal of Applied Mechan-ics, 53, 159–65.
Watson, G. A. (1980), Approximation Theory and Numerical Methods, John Wiley & Sons, NY.
Wei, H.-L. & Billings, S. A. (2002), Identification of time-varying systems using multiresolution wavelet models, In-ternational Journal of Systems Science, 33(15), 1217–28. Wu, C. L., Yang, Y. S. & Loh, C. H. (2006), Dynamic gravity
load collapse of non-ductile RC frames I: experimental ap-proach, Proceedings of the 8th U.S. National Conference on Earthquake Engineering, paper no. 750.
Zou, R., Wang, H. & Chon, K. H. (2003), A robust time-varying identification algorithm using basis functions, An-nals of Biomedical Engineering, 31, 840–53.
APPENDIX Nomenclature
a(t)= acceleration
ag(t)= base excitation input
ai= coefficient vector corresponding to pi
an(t)= residual error in TVARX
c(t)= damping coefficient, function of time d= support of weight function
f(t− i) = input at time t − i t
fn= instantaneous natural frequency (Hz)
(I, J)= order of TVARX k(t)= stiffness, function of time
¯li= the number of nodal points for φi(t)
ˆlj= the number of nodal points for θj(t)
m(t)= mass, function of time
Ni= the highest order of polynomial in pi
¯
Nj= the highest order of polynomial in ¯pj
¯
N= Niand ¯Njfor all i and j
pi= a vector of polynomial basis functions for
φi(t)
¯pj = a vector of polynomial basis functions for θj(t)
v(t) = velocity
W(t, tl)= a weight function
x(t)= displacement
y (t− i) = measured response at time t − it ωn(t)= instantaneous natural frequency (rad/
second)
ω1a= instantaneous natural frequency
identi-fied by using acceleration responses ω1v= instantaneous natural frequency
identi-fied by using velocity responses ξ(t) = instantaneous damping ratio
ξ1a= instantaneous damping ratio identified by
using acceleration responses
ξ1v= instantaneous damping ratio identified by
using velocity responses
φi(t),θj(t)= coefficient functions in TVARX
¯
φi k= true values of φi(tk)
¯
θj k= true values of θj(tk)
˜
ϕi(t)= a vector of shape functions for φi(t)
˜θj(t)= a vector of shape functions for θj(t)
μ = means of relative error in identifying in-stantaneous modal parameters
¯
μ = an error index defined by Equation (46) σ = variances of relative error in identifying
